How to update/overwrite an existing rig with Rigify
Looks like I've found my own answer right after posting this question :)

I had moved my metarig to another layer and when pressing 'Generate Rig' the layer with the rig itself was not active. So make sure to have both the metarig and the rig active when pressing 'Generate Rig'.

'Generate rig' now correctly overwrites my existing rig, keeping the parenting and skinning intact. Only my pose position gets reset, but that's not a problem.